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Switched-Capacitor Circuits
Published in Tertulien Ndjountche, CMOS Analog Integrated Circuits, 2017
The frequency responses of a discrete-time filter, an anti-aliasing filter and the overall system are shown in Figure 8.3. The discrete-time filter has a low-pass characteristic with a cutoff frequency fc and a stopband attenuation As. Note that the spectrum of every discrete-time system is replicated at multiples of the sampling frequency. The transition from the passband to the stopband of the anti-aliasing filter consists of the frequency region located between fs/2 and fs – fc. A lowpass filter prototype is characterized by a cutoff (or passband) frequency, a stopband frequency, a maximum attenuation (or ripple) in the passband, and a minimum attenuation in the stopband. Depending on the type of application, the filter transfer function can be approximated by functions known as the Butterworth, Bessel, Chebyshev, or elliptic response, each of which has its own advantages or disadvantages. The Butterworth filter exhibits the flattest passband and lowest attenuation in the stopband. The Bessel filter has a more gradual roll-off and features a linear phase response, resulting in a constant time delay over a wide range of frequencies through the passband. The Chebyshev filter has a steeper roll-off near the cutoff fre quency and ripples in the passband. The elliptic filter has the steepest roll-off and equal ripples in both passband and stopband.
Introduction
Published in S. A. Pactitis, Active Filters, 2018
Just as the Butterworth filter is the best approximation to the ideal of “perfect flatness of the amplitude response” in the filter passband, so the Bessel filter provides the best approximation to the ideal of “perfect flatness of the group delay” in the passband, because it has a maximally flat group delay response. However, this applies only to low-pass filters because high-pass and band-pass Bessel filters do not have the linear-phase property. Figure 1.11 compares the amplitude (a) and phase response of a Bessel filter with that of a Butterworth filter of the same order. Bessel step response is plotted in Figure 1.12 for various values of n.
The Approximation Problem
Published in T. Deliyannis, Yichuang Sun, J.K. Fidler, Continuous-Time Active Filter Design, 2019
T. Deliyannis, Yichuang Sun, J.K. Fidler
In Fig. 2.5 the amplitude, and in Fig. 2.13 the phase response, of the third-order Bessel filter are shown along with the corresponding responses of the Butterworth and the 1 dB ripple Chebyshev filters of the same order. It can be seen that, from the selectivity point of view, the Bessel filter is at a disadvantage, but its phase response, as far as linearity is concerned, is by far superior—particularly when compared to the Chebyshev phase response.
A Band Rejection Filter of High Current Radio Frequency Ion Source for Neutral Beam Injector
Published in Fusion Science and Technology, 2021
Wei Liu, Qinglong Cui, Sheng Liu, Lizhen Liang, Yuanzhe Zhao, Shihua Song
The “normalization” method is used in the improved BRF, which is based on a Butterworth filter. Figure 1 gives the comparison of frequency response of the different LC filter circuits: Butterworth filter, Bessel filter, Cauer filter, and Chebyshev filter. The Butterworth filter shows the maximally flat in the passband and stopband, the closer the response is to the brick-wall model. The Chebyshev filter maximizes the transition-band cutoff rate at the cost of introducing passband ripples, which is suitable for situations with high attenuation. Compared with the Chebyshev filter, the Cauer filter carries the approach one step further by trading ripples in both the passband and the stopband for an even sharper characteristic in the transition band. The Bessel filter contains a nearly linear phase characteristic within the passband, at the price of a less-sharp magnitude characteristic in the transition band. Considering the cutoff rate in the stopband, the flat in the passband, and the phase response, the Butterworth filter is the promising choice for our system requirement.10